Recall we measure angles in terms of degrees or radians: 360 = 2π radians

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1 Review: trigonometry 8 Review of trigonometry 8.1 Definition of sine and cosine Recall we measure angles in terms of degrees or radians: 360 = 2π radians 8.2 Basic facts: 1. Te point (cos(t), sin(t)) lies on te unit circle wit center (0, 0), so (cos(t)) 2 + (sin(t)) 2 = From te picture cos( π 2 t ) = sin(t). 3. For any angle t, if we add 2π (radians) more, be ave done completely around, so: sin(t + 2π) = sin(t), and cos(t + 2π) = cos(t). We say sin and cos are periodic wit period 2π. 4. cos is an even function, and sin is an odd function. 5. Combining facts 2 and 4, we get cos( t π 2 ) = cos( π 2 t ) = sin(t) Terefore, te grap of sin is rigtward sift of te grap of cos by te amount π Te tangent, secant and cosecant functions are defined as: tan(t) = sin(t) cos(t), sec(t) = 1 cos(t), and cosec(t) = 1 sin(t). Te tangent function tan(t) measures te slope of te angle t, and it is an odd function and periodic wit period π (180 degrees).

2 7. Two important trigonometric identities wic we use later to compute te derivative of te sin and cos functions are: sin ( A + B ) = sin ( A ) cos ( B ) + cos ( A ) sin ( B ) cos ( A + B ) = cos ( A ) cos ( B ) sin ( A ) sin ( B ) 9 Inverse trigonometic functions 9.1 Intervals of real numbers If a < b are two real numbers te set of numbers between a and b is called te interval between a and b. To be more precise, we use te notations: [ a, b ] = { x R a x b } te closed interval between a and b ( a, b ) = { x R a < x < b } te open interval between a and b [ a, b ) = { x R a x < b } te alf open/closed interval between a and b ( a, b ] = { x R a < x b } te alf open/closed interval between a and b for te 4 types of intervals of numbers between a and b. We can also allow te numbers a and b to be infinity: (, b] = { x R x b }, etc, (a, ) = { x R a < x}

3 Te usual domains for te functions cos and sin is te entire set of (real) numbers R. Te range of bot cos, and sin is: range = C = { 1 y 1 } = [ 1, 1]. But, cos and sin are not one-to-one functions on R. If we restrict te domain of sin to be te interval D = [ π 2, π 2 ], were sin is an increasing function, ten [ π 2, π 2 ] sin [ 1, 1 ] is a one-to-one and onto function. Te inverse function is called arcsin. arcsin [ 1, 1 ] [ π 2, π 2 ] 9.2 arcsin graps of sin (red) and arcsin (blue) arcsin(x) sin(x)

4 9.3 arccos For cos, we restrict te domain to te interval [ 0, π ]. Te inverse arccos function is arccos: [ 1, 1 ] [ 0, π ] graps of cos (red) and arccos (blue) arccos(x) cos(x) arctan For tan, we restrict te domain to te open interval ( π 2, π 2 ). Te arctan inverse function is arctan: (, ) ( π 2, π 2 ) graps of tan (red) and arctan (blue) tan(x) arctan(x)

5 14 Some trigonometric inequalities and te its tey yield Consider a circular sector wit center O and radius 1. Let A, B, E be points on te circular sector as sown. E 2 B D O C A Ten: lengt(bc) = sin(), lengt(arc(ab)) =, and lengt(ad) = tan() st inequalties and te it 0 sin() = 1. For 0 < < π 4, so 2 is at most π 2 (90 degrees), we ave: lengt(bc) < lengt(ab) < lengt(arc(ab)), so sin() < lengt(ab) < and lengt(arc(ab)) < lengt(ad), so < tan(). In summary, we ave: sin() < < tan() We manipulate to: cos() < sin() < 1, valid for 0 < < π 4. Te tree functions cos(), sin(), and 1 are all even, so te inequality is also true for π 4 < < 0. In a picture:

6 Te rule sin() does NOT allow input of = 0. Zero is not in te domain. But, te function sin() is caugt (for 0) between te two functions cos() and 1. We can apply te squeeze teorem to get sin() 0 = nd it involving te function 1 cos() From te 1st picture we ave: lengt(ac) 2 + lengt(bc) 2 = lengt(ab) 2 lengt(arc(ab)) 2, so (1 cos()) 2 + (sin()) 2 2, ten expand to get 2 ( 1 cos() ) 2, and deduce 0 ( 1 cos() ) 2, for 0 < < π 4 Te (odd symmetry) rule ( 1 cos() ) does not allow input = 0.

7 A picture of tis rule (grap in red), wit values of ( 1 cos() ) caugt between 0 (grap in yellow) and 2 (grap in blue), is: As before, we can deduce 0 ( 1 cos() ) = 0. sin() ( 1 cos() ) Tese two its 0 = 1, and 0 = 0 are very important. Tey are used later to compute te slope of tangent lines to te functions sin and cos. 15 Tangent slope of te functions sine and cosine 15.1 Tangent slope to te grap of sine at te point (b, sin(b)). Te tangent slope at te grap point (b, sin(b)) is te it of te difference quotient: sin( b + ) sin( b ). We use te formula for te sine of a sum to get: sin( b + ) sin( b ) = sin(b) cos() + cos(b) sin() sin(b) = sin(b) ( cos() 1 ) + cos(b) sin() sin( b + ) sin( b ) 0 ( = sin(b) ( cos() 1 ) 0 ( cos() 1 ) = sin(b) 0 = sin(b) 0 + cos(b) 1 = cos(b) + cos(b) sin() ) + cos(b) 0 sin() Te tangent slope to te grap of sine at te point (b, sin(b)) is cos(b).

8 15.2 Tangent slope to te grap of cosine at te point (b, cos(b)). Te tangent slope at te grap point (b, cos(b)) is te it of te difference quotient: cos( b + ) cos( b ). We use te formula for te cosine of a sum to get: cos( b + ) cos( b ) = cos(b) cos() sin(b) sin() cos(b) = cos(b) ( cos() 1 ) sin(b) sin() cos( b + ) cos( b ) 0 ( = cos(b) ( cos() 1 ) 0 = cos(b) 0 ( cos() 1 ) sin(b) sin() ) sin(b) 0 sin() Our earlier calculation using te squeeze teorem sowed so, ( cos() 1 ) 0 = 0 and 0 sin() = 1 ; cos( b + ) cos( b ) = cos(b) 0 sin(b) 1 = sin(b) 0 Te tangent slope to te grap of cosine at te point (b, cos(b)) is sin(b).